What is the sum of all the integers between 19 and 77

Think of it this way: Lets add numbers in pairs, starting at the very outer 2 numbers (19 and 77) then go in by one and add the second number and the second to last (20 and 76), then (21 and 75) and so on. The sum of all of these pairs are all the same: 96. How many 96s will we have? Well since we're coming from each end toward the middle adding pairs we will have half the distance between 19 and 77, that is (77-19) / 2 = 29. So we can actually just take 96*29 = 2784. This is the sum of all numbers between 19 and 77